1. Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders Speaker: Michael Schapira Joint work with Shahar Dobzinski & Noam Nisan

2. 2 Talk Structure  Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation of CF auctions using value queries. 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

3. 3 Combinatorial Auctions A set M of items for sale. |M|=m. n bidders, each bidder i has a valuation function v i :2 M ->R +. Common assumptions: Normalization: v i (  )=0 Free disposal: S  T  v i (T) ≥ v i (S) Goal: find a partition S 1,…,S n such that social welfare  v i (S i ) is maximized

4. 4 Combinatorial Auctions Problem 1: finding an optimal allocation is NP- hard. Problem 2: valuation length is exponential in m. Problem 3: how can we be certain that the bidders do not lie ? (incentive compatibility)

5. 5 Combinatorial Auctions We are interested in algorithms that based on the reported valuations {v i } i output an allocation which is an approximation to the optimal social welfare. We require the algorithms to be polynomial in m and n. That is, the algorithms must run in sub- linear (polylogarithmic) time. We explore the achievable approximation factors.

6. 6 Access Models How can we access the input ? One possibility: bidding languages. The “black box” approach: each bidder is represented by an oracle which can answer certain queries.

7. 7 Access Models Common types of queries:  Value: given a bundle S, return v(S).  Demand: given a vector of prices (p 1,…, p m ) return the bundle S that maximizes v(S)-  j  S p j.  General: any possible type of query (the comunication model). Demand queries are strictly more powerful than value queries (Blumrosen-Nisan, Dobzinski-Schapira)

8. 8 Known Results Finding an optimal solution requires exponential communication. Nisan-Segal Finding an O(m 1/2-  )-approximation requires exponential communication. Nisan-Segal. ( this result holds for every possible type of oracle) Using demand oracles, a matching upper bound of O(m 1/2 ) exists (Blumrosen-Nisan). Better results might be obtained by restricting the classes of valuations.

9. 9 The Hierarchy of CF Valuations Complement-Free: v(S  T) ≤ v(S) + v(T). XOS: XOR of ORs of singletons  Example: (A:2 OR B:2) XOR (A:3) Submodular: v(S  T) + v(S  T) ≤ v(S) + v(T).  2-approximation by LLN. GS: (Gross) Substitutes, OXS: OR of XORs of singletons  Solvable in polynomial time (LP and Maximum Weighted Matching respectively) OXS  GS  SM  XOS  CF Lehmann, Lehmann, Nisan

10. 10 Talk Structure Combinatorial Auctions  Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auctions using value queries. 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

11. 11 Intuition We will allow the auctioneer to allocate k duplicates from each item. Each bidder is still interested in at most one copy of each item (so valuations are kept the same). Using the assumption that all valuations are CF, we will find an approximation to the original auction, based on the k-duplicates allocation.

12. 12 The Algorithm – Step 1 Solve the linear relaxation of the problem: Maximize:  i,S x i,S v i (S) Subject To:  For each item j:  i,S|j  S x i,S ≤ 1  For each bidder i:  S x i,S ≤ 1  For each i,S: x i,S ≥ 0 Despite the exponential number of variables, the LP relaxation may still be solved in polynomial time using demand oracles. (Nisan-Segal). OPT*=  i,S x i,S v i (S) is an upper bound for the value of the optimal integral allocation.

13. 13 The Algorithm – Step 2 Use randomized rounding to build a “pre- allocation” S 1,..,S n :  Each item j appears at most k=O(log(m)) times in {S i } i.   i v i (S i ) ≥ OPT*/2. Randomized Rounding: For each bidder i, let S i be the bundle S with probability x i,S, and the empty set with probability 1-  S x i,S.  The expected value of v i (S i ) is  S x i,S v i (S) We use the Chernoff bound to show that such “pre-allocation” is built with high probability.

14. 14 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = .

15. 15 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD

16. 16 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD S 1 1 = {A,B,D}

17. 17 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD

18. 18 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . CE AD S 2 1 = {C,E} S 2 2 = {A,D}

19. 19 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A

20. 20 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . CE A S 3 2 = {C,E} S 3 3 = {A}

21. 21 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A B D

22. 22 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A B DBCE

23. 23 The Algorithm – Step 4 Find the t maximizes  i v i (S i t ) Return the allocation (S 1 t,...,S n t ). All valuations are CF so:  t  i v i (S i t ) =  i  t v i (S i t ) ≥  i v i (S i ) ≥ OPT*/2  For the t that maximizes  i v i (S i t ), it holds that:  i v i (S i t ) ≥ (  i v i (S i ))/k ≥ OPT*/2k = OPT*/O(log(m)). ABD CE AD CE A B DBCE

24. 24 A Communication Lower Bound of 2-  for CF Valuations Theorem: Exponential communication is required for approximating the optimal allocation among CF bidders to any factor less than 2. Proof: A simple reduction from the general case.

25. 25 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions  An incentive compatible O(m 1/2 )- approximation of CF auctions using value queries. 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

26. 26 Incentive Compatibility & VCG Prices We want an algorithm that is truthful (incentive compatible). I.e. we require that the dominant strategy of each of the bidders would be to reveal true information. VCG is the only general technique known for making auctions incentive compatible (if bidders are not single-minded):  Each bidder i pays:  k≠i v k (O -i ) -  k≠i v k (O i ) O i is the optimal allocation, O -i the optimal allocation of the auction without the i’th bidder.

27. 27 Incentive Compatibility & VCG Prices Problem: VCG requires an optimal allocation! Finding an optimal allocation requires exponential communication and is computationally intractable. Approximations do not suffice (Nisan-Ronen).

28. 28 VCG on a Subset of the Range Our solution: limit the set of possible allocations.  We will let each bidder to get at most one item, or we’ll allocate all items to a single bidder. Optimal solution in the set can be found in polynomial time  VCG prices can be computed  incentive compatibility. We still need to prove that we achieve an approximation.

29. 29 The Algorithm Ask each bidder i for v i (M), and for v i (j), for each item j. (We have used only value queries) Construct a bipartite graph and find the maximum weighted matching P.  can be done in polynomial time (Tarjan). 1 2 3 A B Items Bidders v 1 (A) v 3 (B)

30. 30 The Algorithm (Cont.) Let i be the bidder that maximizes v i (M). If v i (M)>|P|  Allocate all items to i. else  Allocate according to P. Let each bidder pay his VCG price (in respect to the restricted set).

31. 31 Proof of the Approximation Ratio Theorem: If all valuations are CF, the algorithm provides an O(m 1/2 )-approximation. Proof: Let OPT=(T 1,..,T k,Q 1,...,Q l ), where for each T i, |T i |>m 1/2, and for each Q i, |Q i |≤m 1/2. |OPT|=  i v i (T i ) +  i v i (Q i ) Case 1:  i v i (T i ) >  i v i (Q i ) (“large” bundles contribute most of the social welfare)   i v i (T i ) > |OPT|/2 At most m 1/2 bidders get at least m 1/2 items in OPT.  For the bidder i the bidder i that maximizes v i (M), v i (M) > |OPT|/2m 1/2. Case 2:  i v i (Q i ) ≥  i v i (T i ) (“small” bundles contribute most of the social welfare)   i v i (Q i ) ≥ |OPT|/2 For each bidder i, there is an item c i, such that: v i (c i ) > v i (Q i ) / m 1/2. (The CF property ensures that the sum of the values is larger than the value of the whole bundle) {c i } i is an allocation which assigns at most one item to each bidder: |P| ≥  i v i (c i ) ≥ |OPT|/2m 1/2.

32. 32 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auction  2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

33. 33 Definition of XOS XOS: XOR of ORs of Singletons. Singleton valuation (x:p)  v(S) = p x  S 0 otherwise Example: (A:2 OR B:2) XOR (A:3)

34. 34 XOS Properties The strongest bidding language syntactically restricted to represent only complement-free valuations. Can describe all submodular valuations (and also some non-submodular valuations) Can describe interesting NPC problems (Max-k-Cover, SAT).

35. 35 Supporting Prices Definition: p 1,…,p m supports the bundle S in v if:  v(S) =  j  S p j  v(T) ≥  j  T p j for all T  S Claim: a valuation is XOS iff every bundle S has supporting prices. Proof:   There is a clause that maximizes the value of a bundle S. The prices in this clause are the supporting prices.   Take the prices of each bundle, and build a clause.

36. 36 Algorithm-Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}.

37. 37 Algorithm-Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}. Price vector: p 1 =(1,1,1,0,0) v 2 : (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2) Bidder 2 gets his demand: {C}

38. 38 Algorithm-Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}. Price vector: p 1 =(1,1,1,0,0) v 2 : (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2) Bidder 2 gets his demand: {C} Price vector: p 2 =(1,1,9,0,0) v 3 : (C:10 OR D:1 OR E:2) Bidder 3 gets his demand: {C,D,E} Final allocation: {A,B} to bidder 1, {C,D,E} to bidder 3.

39. 39 The Algorithm Input: n bidders, for each we are given a demand oracle and a supporting prices oracle. Init: p 1 =…=p m =0. For each bidder i=1..n  Let S i be the demand of the i’th bidder at prices p 1,…,p m.  For all i’ < i take away from S i’ any items from S i.  Let q 1,…,q m be the supporting prices for S i in v i.  For all j  S i update p j = q j.

40. 40 Proof To prove the approximation ratio, we will need these two simple lemmas: Lemma: The total social welfare generated by the algorithm is at least  p j. Lemma: The optimal social welfare is at most 2  p j.

41. 41 Proof – Lemma 1 Lemma: The total social welfare generated by the algorithm is at least  p j. Proof: Each bidder i got a bundle T i at stage i. At the end of the algorithm, he holds A i  T i. The supporting prices guarantee that: v i (A i ) ≥  j  Ai p j

42. 42 Proof – Lemma 2 Lemma: The optimal social welfare is at most 2  p j. Proof: Let O 1,...,O n be the optimal allocation. Let p i,j be the price of the j’th item at the i’th stage. Each bidder i ask for the bundle that maximizes his demand at the i’th stage: v i (O i )-  j  Oi p i,j ≤  j p i,j –  j p (i-1),j Since the prices are non-decreasing: v i (O i )-  j  Oi p n,j ≤  j p i,j –  j p (i-1),j Summing up on both sides:  i v i (O i )-  i  j  Oi p n,j ≤  i (  j p i,j –  j p (i-1),j )  i v i (O i )-  j p n,j ≤  j p n,j  i v i (O i ) ≤ 2  j p n,j

43. 43 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation of CF auctions using value queries. 2-approximation for XOS auctions  A lower bound of e/(e-1)-  for XOS auctions

44. 44 XOS Lower Bounds: We show two lower bounds:  A communication lower bound of e/(e-1)-  for the “black box” approach.  An NP-Hardness result of e/(e-1)-  for the case that the input is given in XOS format (bidding language). We now prove the second of these results.

45. 45 Max-k-Cover We will show a polynomial time reduction from Max-k-Cover. Max-k-Cover definition:  Input: a set of |M|=m items, t subsets S i  M, an integer k.  Goal: Find k subsets such that the number of items in their union, |  S i |, is maximized. Theorem: approximating Max-k-Cover within a factor of e/(e-1) is NP-hard (Feige).

46. 46 The Reduction ABC DEF v 1 : (A:1 OR D:1) XOR (C:1 OR F:1) XOR (D:1 OR E:1 OR F:1) v k : (A:1 OR D:1) XOR (C:1 OR F:1) XOR (D:1 OR E:1 OR F:1) Every solution to Max-k-Cover implies an allocation with the same value. Every allocation implies a solution to Max-k-Cover with at least that value.  Same approximation lower bound. A matching communication lower bound exists. Max-k-Cover InstanceXOS Auction with k bidders

47. 47 Open Questions – Narrowing the Gaps Valuation Class Value queriesDemand queries General communication General≤ m/(log 1/2 m) (Holzman, Kfir- Dahav, Monderer, Tennenholz) ≥ m/(logm) (Nisan-Segal, Dobzinki-Schapira) ≤ m 1/2 (Blumrosen- Nisan) ≥ m 1/2 (Nisan-Segal) CF≤ m 1/2 ≤ log(m) ≥ 2≥ 2 XOS≤ 2 ≥ e/(e-1) SM≤ 2 (Lehmann,Lehmann,Nisan) ≥ e/(e-1 )(new: Khot, Lipton,Markakis, Mehta) ≥ 1+1/(2m) (Nisan-Segal) GS1 (Bertelsen, Lehmann)